MTHSC 102 Section 4.2 – Relative and
Absolute Extreme Points
Kevin James
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Definition (Extreme Points)
Suppose that f (x) is a function defined on an interval I .
1
We say that f attains a relative maximum value of f (a) at
x = a if there is some interval (b, c) with b < a < c and such
that for all x ∈ (b, c), f (x) ≤ f (a).
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Definition (Extreme Points)
Suppose that f (x) is a function defined on an interval I .
1
We say that f attains a relative maximum value of f (a) at
x = a if there is some interval (b, c) with b < a < c and such
that for all x ∈ (b, c), f (x) ≤ f (a).
2
We say that f attains a relative minimum value of f (a) at
x = a if there is some interval (b, c) with b < a < c and such
that for all x ∈ (b, c), f (x) ≥ f (a).
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Definition (Extreme Points)
Suppose that f (x) is a function defined on an interval I .
1
We say that f attains a relative maximum value of f (a) at
x = a if there is some interval (b, c) with b < a < c and such
that for all x ∈ (b, c), f (x) ≤ f (a).
2
We say that f attains a relative minimum value of f (a) at
x = a if there is some interval (b, c) with b < a < c and such
that for all x ∈ (b, c), f (x) ≥ f (a).
3
We say that f attains an absolute maximum value on I of
f (a) at x = a if for all x ∈ I , f (x) ≤ f (a).
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Definition (Extreme Points)
Suppose that f (x) is a function defined on an interval I .
1
We say that f attains a relative maximum value of f (a) at
x = a if there is some interval (b, c) with b < a < c and such
that for all x ∈ (b, c), f (x) ≤ f (a).
2
We say that f attains a relative minimum value of f (a) at
x = a if there is some interval (b, c) with b < a < c and such
that for all x ∈ (b, c), f (x) ≥ f (a).
3
We say that f attains an absolute maximum value on I of
f (a) at x = a if for all x ∈ I , f (x) ≤ f (a).
4
We say that f attains an absolute minimum value on I of
f (a) at x = a if for all x ∈ I , f (x) ≥ f (a).
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Definition (Extreme Points)
Suppose that f (x) is a function defined on an interval I .
1
We say that f attains a relative maximum value of f (a) at
x = a if there is some interval (b, c) with b < a < c and such
that for all x ∈ (b, c), f (x) ≤ f (a).
2
We say that f attains a relative minimum value of f (a) at
x = a if there is some interval (b, c) with b < a < c and such
that for all x ∈ (b, c), f (x) ≥ f (a).
3
We say that f attains an absolute maximum value on I of
f (a) at x = a if for all x ∈ I , f (x) ≤ f (a).
4
We say that f attains an absolute minimum value on I of
f (a) at x = a if for all x ∈ I , f (x) ≥ f (a).
We refer to these values as extreme values of f .
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Fact
If f is a smooth continuous function and if f attains an extreme
value at x = a, then the derivative f ′ crosses the input axis at
x = a and thus f ′ (a) = 0.
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Example
Consider the function
(
x 3 − 12x + 17
f (x) =
x2 + 5
Kevin James
if x ≤ 1,
if x > 1.
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Example
Consider the function
(
x 3 − 12x + 17
f (x) =
x2 + 5
if x ≤ 1,
if x > 1.
The graph of this function and its derivative are
Find all relative and absolute extreme points of f in the interval
[−4, 4].
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Note
Relative extrema do not occur if the derivative touches the input
axis but does not cross it. That is, f ′ (a) may be zero even when f
does not attain a relative extreme at x = a.
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Note
Relative extrema do not occur if the derivative touches the input
axis but does not cross it. That is, f ′ (a) may be zero even when f
does not attain a relative extreme at x = a.
Note
For a function f with input x, a relative extremum can occur at
x = c only if f (c) exists. Furthermore,
1
A relative extremum exists where f ′ (c) = 0 and the graph of
f ′ (x) crosses the input axis at x = c.
2
A relative extremum can exist where f (x) exists but f ′ (x)
does not.
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Finding Extrema
To find the relative maxima and minima of a function f ,
1
Determine the input values for which f ′ = 0 of f ′ is undefined.
2
Examine a graph of f to determine which of these input
values correspond to relative maxima or relative minima.
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Finding Extrema
To find the relative maxima and minima of a function f ,
1
Determine the input values for which f ′ = 0 of f ′ is undefined.
2
Examine a graph of f to determine which of these input
values correspond to relative maxima or relative minima.
To find the absolute maximum and minimum of a function f on an
interval [a, b].
1
Find all relative extrema of f in the interval (as above).
2
Compare the relative extreme values in the interval and f (a)
and f (b). The largest value is the absolute maximum value
and the smallest value is the absoluet minimum.
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Finding Extrema
To find the relative maxima and minima of a function f ,
1
Determine the input values for which f ′ = 0 of f ′ is undefined.
2
Examine a graph of f to determine which of these input
values correspond to relative maxima or relative minima.
To find the absolute maximum and minimum of a function f on an
interval [a, b].
1
Find all relative extrema of f in the interval (as above).
2
Compare the relative extreme values in the interval and f (a)
and f (b). The largest value is the absolute maximum value
and the smallest value is the absoluet minimum.
To find the absolute maximum and minimum of a continuous
function f without a specified interval.
1
Find all relative extrema of f .
2
Determine the end behavior of the function in both directions.
The absolute extrema either do not exist or are among the
relative extrema.
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin
Example
The number of consumer complaints to the US Department of
Transportation about baggage on US airlines between 1989 and
2000 can be modeled by the function
B(x) = 55.15x 2 − 524.09x + 1768.65 complaints,
where x is the number of years after 1989.
1
The graph of the function is
2
Find the relative and absolute maxima and minima on the
interval 0 ≤ x ≤ 11.
Kevin James
MTHSC 102 Section 4.2 – Relative and Absolute Extreme Poin